The class of functions of bounded n-th variation, denoted by BVn[a,b], was introduced by M.T. Popoviciu in 1933. In 1979 A.M. Russell in [6] proved the Jordan-type decomposition theorem for functions from this class, then, applying this result, showed that each BVn[a,b], with a suitable norm, is a Banach space. For n=0 and n=1 the above facts are well known classical results (cf., for instance, [5], [7]).
However, the proofs given by A . M . Russell for n≥2, based on some properties of divided differences, are rather complicated. The aim of this note, is to give an essentially simpler arguments both, for the Jordan-type decomposition theorem as well as for the completness of the space BVn[a,b]. In our proofs we apply the Popoviciu theorem and the fact that for every positive integer n≥2, a function f is n-convex iff the derivative f' is (n-1)-convex.
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Vol. 15 (2001)
Published: 2001-09-28
10.1515/amsil
English